Electromagnetism and Optics Charudatt Kadolkar · Electromagnetism and Optics Charudatt Kadolkar...

Physics II

Electromagnetism and Optics

Charudatt Kadolkar

Indian Institute of Technology Guwahati

Jan 2009

Physics II Syllabus

Vector Calculus: Gradient, Divergence and Curl. Line, Surface, and Volume integrals. Gauss'sdivergence theorem and Stokes' theorem in Cartesian, Spherical polar, and Cylindrical polar coordinates.Dirac Delta function.

Electrodynamics: Coulomb's law and Electrostatic �eld, Fields of continuous charge distributions.Gauss's law and its applications. Electrostatic Potential. Work and Energy. Conductors. Capacitors.Laplace's equation. Method of images. Dielectrics. Polarization. Bound Charges. Energy in dielectrics.Boundary conditions.

Lorentz force. BiotSavart and Ampere's laws and their applications. Vector Potential. Force and torqueon a magnetic dipole. Magnetic materials. Magnetization, Bound currents. Boundary conditions.Motional EMF, Ohm's law. Faraday's law. Lenz's law. Self and Mutual inductance. Energy stored inmagnetic �eld. Maxwell's equations.

Optics: Huygens' principle. Young's experiment. Superposition of waves. Concepts of coherencesources. Interference by division of wavefront. Fresnel's biprism, Phase change on re�ection. Lloyd'smirror. Interference by division of amplitude. Parallel �lm. Film of varying thickness. Colours of thin�lms. Newton's rings. The Michelson interferometer. Fraunhofer di�raction. Single slit, double slit andN-slit patterns. The di�raction grating.

Reading Material

I Textbooks

1. D. J. Gri�ths, Introduction to Electrodynamics, Prentice-Hall (1995).2. F. A. Jenkins and H. E. White, Fundamental of Optics, McGraw-Hill,

(1981).

I References:

1. Feynman, Leighton, and Sands, The Feynman Lectures on Physics,Vol. II, Norosa Publishing House (1998).

2. I. S. Grant and W. R. Phillips, Electromagnetism, John Wiley, (1990).3. E. Hecht, Optics, Addison-Wesley, (1987).

Vector Analysis

MA101: Mathematics IVector functions of one variable and their derivatives. Functions of severalvariables, partial derivatives, chain rule, gradient and directional derivative.Tangent planes and normals. Maxima, minima, saddle points, Lagrangemultipliers, exact di�erentials.Repeated and multiple integrals with application to volume, surface area,moments of inertia. Change of variables. Vector �elds, line and surfaceintegrals. Green's, Gauss' and Stokes' theorems and their applications.

Scalar And Vector Quantities

Physical Quantity

Observables, which can be measured and represented by numbers, are calledphysical quantities.

Some Examples

I mass of a body,

I volume of a body,

I temperature at a point etc.

Some More Examples

I position of a particle,

I electric �eld at a point

I size of a cuboid,

I thermodynamic state (ρ,P,T )

Scalar quantities have magnitude and vector quantities have magnitude as wellas direction.

Vectors are represented by three real numbers.

Is a physical quantity represented by three numbers, a vector?


Physical Quantity


Some Examples

I mass of a body,

I volume of a body,


Some More Examples



I size of a cuboid,






Physical Quantity


Some Examples

I mass of a body,

I volume of a body,


Some More Examples



I size of a cuboid,






Physical Quantity


Some Examples

I mass of a body,

I volume of a body,


Some More Examples



I size of a cuboid,






Physical Quantity


Some Examples

I mass of a body,

I volume of a body,


Some More Examples



I size of a cuboid,





Alternate De�nition: Vector Quantities

Two observers with di�erent coordinate systems are shown in the �gure.

X

Y

P

Α

x

y

X’

Y’

x’

y’

The coordinates of a point P

x ′ = x cosα+ y sinα

y ′ = −x sinα+ y cosα[x ′

y ′

]=

[cosα sinα− sinα cosα

] [x

y

]

X

Y

Α

A

X’

Y’

Components of A in two frames arerelated by[

A′x

A′y

]=


] [Ax

Ay

]


Two observers with di�erent coordinate systems are shown in the �gure.

X

Y

P

Α

x

y

X’

Y’

x’

y’

The coordinates of a point P

x ′ = x cosα+ y sinα

y ′ = −x sinα+ y cosα[x ′

y ′

]=


] [x

y

]

X

Y

Α

A

X’

Y’

Components of A in two frames arerelated by[

A′x

A′y

]=


] [Ax

Ay

]


De�nition

Suppose a physical quantity A is represented by n components in n-dimensionalspace. If, under a coordinate transformation, the components of A transform inthe same way as coordinates, then the physical quantity A is said to be a vector

quantity.

Scalar quantities donot depend on the coordinate transformations.

size of cuboid, a state of a thermodynamic system are not a vector quantitiesbut are collections of three scalar quantities.


De�nition


quantity.




De�nition


quantity.



Scalar And Vector Functions of One Variable

Typically, in Physics, scalar and vector quantities are functions of time.Examples are:

I Temperature of a body

I Amount of Radioactive material etc.

I Position of a particle

I Forces on a particle

We are quite familiar with scalar functions and calculus of such functions.Calculus of vector valued functions is also familiar to us.

Example

Suppose r : R→ R3 is position of a particle as a function of time. Letr (t) = x(t)x + y(t)y + z(t)z. Then, velocity of the particle is given by

v(t) =d

dtr (t) =

dx

dt(t)x +

dy

dt(t)y +

dz

dt(t)z.

Scalar And Vector Functions of One Variable

Typically, in Physics, scalar and vector quantities are functions of time.Examples are:

I Temperature of a body

I Amount of Radioactive material etc.

I Position of a particle

I Forces on a particle

We are quite familiar with scalar functions and calculus of such functions.Calculus of vector valued functions is also familiar to us.

Example

Suppose r : R→ R3 is position of a particle as a function of time. Letr (t) = x(t)x + y(t)y + z(t)z. Then, velocity of the particle is given by

v(t) =d

dtr (t) =

dx

dt(t)x +

dy

dt(t)y +

dz

dt(t)z.

Scalar and Vector Fields

While describing extended objects that �ll space or regions of space, we use�elds, that is, with each point of the space, we associate a vector or scalarproperty.

Examples are:

I Air temperature across the Earth,

I Height of a location from sea level,

I Density of a �uid,

I Force Fields (Gravitational, Electrostatic),

I Potential energy of a force �eld etc.

Mathematically, these are functions from Rm to Rn, with m > 1. If n = 1,these functions are called scalar �elds otherwise are called vector �elds.



Examples are:









Examples are:







Scalar Fields (2D)

x

y

z

f (x , y) = x2 + y2

Scalar Fields (2D)

De�nition

Let f : Rm → R. Then, S = {a ∈ Rm|f (a) = c} is called a level set.

If m = 2, level sets are called level curves, otherwise level surfaces.

x

y

z

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

Scalar Fields (2D)

De�nition



x

y

z

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

Scalar Fields (2D)

De�nition



x

y

z

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

Scalar Fields (2D)

De�nition



x

y

z

-2 -1 0 1 2-2

-1

0

1

2

0

4

Scalar Fields (2D)

De�nition



x

y

z

-2 -1 0 1 2-2

-1

0

1

2

0

4

Scalar Fields (2D)

De�nition



x

y

z

-2 -1 0 1 2-2

-1

0

1

2

0

4

Scalar Fields (3D)

Two identical point charges at (1, 0, 0) and (−1, 0, 0). The potential in xy-plane

V (x , y , z = 0) =Q

4πε0

1√(x − 1)2 + y2

+1√

(x + 1)2 + y2

-2

0

2

-2

-1

01

2

0

2

4

6-2

-1

01

-3 -2 -1 0 1 2 3-2

-1

0

1

2

Scalar Fields (3D)


V (x , y , z = 0) =Q

4πε0

1√(x − 1)2 + y2

+1√

(x + 1)2 + y2

-2

0

2

-2

-1

01

2

0

2

4

6-2

-1

01

-3 -2 -1 0 1 2 3-2

-1

0

1

2

Scalar Fields (3D)


V (x , y , z = 0) =Q

4πε0

1√(x − 1)2 + y2

+1√

(x + 1)2 + y2

-2

0

2

-2

-1

01

2

0

2

4

6-2

-1

01

-3 -2 -1 0 1 2 3-2

-1

0

1

2

Scalar Fields (3D)

Level surfaces

f (x , y , z) = x2 + y2 − z2 = c

Vector Fields (2D)

f (x , y) = x x + y y

Vector Fields (2D)

-2 -1 1 2

-2

-1

1

2

f (x , y) = −y x + x y

Vector Fields (2D)

Two opposite point charges at (1, 0, 0) and (−1, 0, 0). The electric �eld inxy-plane

E(x , y) =Q

4πε0

((x − 1) x + y y(

(x − 1)2 + y2)3/2

− (x + 1) x + y y((x + 1)2 + y2

)3/2

)

Directional Derivative and Partial Derivatives

x

y

z

a t

f (x , y) = x2 + y2

a = (1, 0)t = (1, 1)/

√2

Consider function f (x , y) = x2 + y2


x

y

z

a t

f (x , y) = x2 + y2

a = (1, 0)t = (1, 1)/

√2

Equation of line: y = x − 1.Or parametric equation: x(h) = 1 + 1√

2h and y(h) = 1√

2h.

Or (x , y) = a + ht.


x

y

z

a t

f (x , y) = x2 + y2

a = (1, 0)t = (1, 1)/

√2

-2 -1.5 -1 -0.5 0.5h

fHhL

Write: f (h) = f (x(h), y(h)) = f (a + ht) = 1 +√2h + h2

f ′ (a; t) =df

dh(h = 0) =

√2


x

y

z

a t

f (x , y) = x2 + y2

a = (1, 0)t = (1, 1)/

√2

-2 -1.5 -1 -0.5 0.5h

fHhL

Equation of the tangent to f (h): 1 +√2h

Equation of the tangent to f (x , y): x = 1 + h/√2, y = h/

√2, z =

√2h


I Let f (x , y) = x2 + y2. Let a = (1, 0) and t = (1, 1)/√2 .

I Line parallel to t and passing through a:

x = 1 +1√2h and y =

1√2h.

with (x , y) = a at h = 0.

I Restriction of function f along this line:

f (h) = f (x(h), y(h)) = 1 +√2h + h2

I Directional derivative

f ′ (a; t) =df

dh(h = 0) =

√2

I Equation of the tangent to f (h): y = 1 +√2h.

I Equation of the tangent to f (x , y): x = 1 + t/√2, y = t/

√2, z =

√2t


De�nition

Let f : R3 → R be a scalar �eld. Let a ∈ R3 be a point in space. Supposet ∈ R3 be a unit vector. Then

limh→0

f (a + ht)− f (a)

h

is called directional derivative of f at a in the direction of t and is denoted byf ′(a; t). (If the limit exists.)

The directional derivative in any one of the coordinate unit vectors, that is,either x, y or z, is also called partial derivative. Partial derivatives are usuallydenoted by

∂

∂xf (a),

∂

∂yf (a) or

∂

∂zf (a).

Example

In previous example: f (x , y) = x2 + y2 and a = (1, 0) then

∂

∂xf (a) = 2,

∂

∂yf (a) = 0.


De�nition


limh→0

f (a + ht)− f (a)

h



∂

∂xf (a),

∂

∂yf (a) or

∂

∂zf (a).

Example


∂

∂xf (a) = 2,

∂

∂yf (a) = 0.


De�nition


limh→0

f (a + ht)− f (a)

h



∂

∂xf (a),

∂

∂yf (a) or

∂

∂zf (a).

Example


∂

∂xf (a) = 2,

∂

∂yf (a) = 0.

Di�erentiability

Notion of di�erentiability

If a function f : R→ R is di�erentiable at a then

f (x) ≈ f (a) + f ′(a)(x − a).

In other words, we can approximate the graph of f by a straight line near a.

Example

In earlier example,f (x) = 1 +

√2x + x2 and a = 0.

Thus f (0) = 1 and f ′(0) =√2 Thus

a line y = 1 +√2x is a good

aprroximation to f at 0.

-2 -1.5 -1 -0.5 0.5h

fHhL

A function is di�erentiable only if the �rst order Taylor expansion is valid.

Di�erentiability



f (x) ≈ f (a) + f ′(a)(x − a).


Example


√2x + x2 and a = 0.

Thus f (0) = 1 and f ′(0) =√2 Thus



-2 -1.5 -1 -0.5 0.5h

fHhL


Di�erentiability



f (x) ≈ f (a) + f ′(a)(x − a).


Example


√2x + x2 and a = 0.

Thus f (0) = 1 and f ′(0) =√2 Thus



-2 -1.5 -1 -0.5 0.5h

fHhL


Total Derivative

Di�erentiability for functions of two variables

If a function of two variables is to be di�erentiable, one must be approximateits graph by a plane rather than a line.

Example

Consider f (x , y) = x2 + y2 anda = (1, 0). Figure shows a planegiven by a functionz(x , y) = 1 + 2(x − 1). Here iscomparison

f (1.1, 0) = 1.21 z(1.1, 0) = 1.20,

f (1, 0.1) = 1.01 z(1, 0.1) = 1.00.

x

y

zf= x2

+ y2

z= 2 x-1

Total Derivative

De�nition

Let f : Rm → R be a scalar �eld. Let a ∈ Rm be a point in space. f is said tobe di�erentiable at a if there exists a linear function Ta : Rm → R such that

f (a + v) = f (a) + Ta (v) + |v|Ra(v).

where Ra(v)→ 0 as v→ 0. The linear transformation Ta is called thederivative of f at a.

Total Derivative

De�nition


f (a + v) = f (a) + Ta (v) + |v|Ra(v).


Linear Transformation m = 2

Let v = (x , y) ∈ R2. A linear transformation T : R2 → R can always be writtenas

T (v) = txx + tyy

where tx , ty ∈ R. That is, one can always �nd a vector t = (tx , ty ) such that

T (v) = t · v

Total Derivative

De�nition


f (a + v) = f (a) + Ta (v) + |v|Ra(v).


Linear Transformation

Let v ∈ Rm. A linear transformation T : Rm → R can always be written as

T (v) = t · v

where t ∈ Rm is some constant vector.

Total Derivative

De�nition


f (a + v) = f (a) + Ta (v) + |v|Ra(v).


Example

f (x , y) = x2 + y2, a = (1, 0), v = (x , y). Then

f (a + v) = f (x + 1, y) = (x + 1)2 + y2 = 1 + 2x + 0y + x2 + y2

= 1 + (2, 0) · v + |v|2

Thus Ta (v) = (2, 0) · v. And remember ∂∂x

f (a) = 2, ∂∂y

f (a) = 0.

Gradient

De�nition


f (a + v) = f (a) + Ta (v) + |v|Ra(v).


Let Ta = t · v where t ∈ Rm , the vector t is called thegradient of f at a and isdenoted by ∇f (a).

Properties of Gradient

Theorem

If f : R3 → R is a di�erentiable at a with total derivative Ta, then all partial

derivatives exist and

Ta (v) =

(∂

∂xf (a),

∂

∂yf (a),

∂

∂zf (a)

)· v

that is

∇f (a) =

(∂

∂xf (a),

∂

∂yf (a),

∂

∂zf (a)

).

Example

f (x , y) = x2 + y2, a = (1, 0), v = (x , y). Then Ta (v) = (2, 0) · v. And∂∂x

f (a) = 2, ∂∂y

f (a) = 0.

The converse of the previous theorem does hold.

Example

f (x , y) = xy2

x2+y2. Show that all partial derivatives of f exist at (0, 0) but total

derivative does not exist.


Theorem



Ta (v) =

(∂

∂xf (a),

∂

∂yf (a),

∂

∂zf (a)

)· v

that is

∇f (a) =

(∂

∂xf (a),

∂

∂yf (a),

∂

∂zf (a)

).

Example


f (a) = 2, ∂∂y

f (a) = 0.


Example

f (x , y) = xy2




Theorem



Ta (v) =

(∂

∂xf (a),

∂

∂yf (a),

∂

∂zf (a)

)· v

that is

∇f (a) =

(∂

∂xf (a),

∂

∂yf (a),

∂

∂zf (a)

).

Example


f (a) = 2, ∂∂y

f (a) = 0.


Example

f (x , y) = xy2




Theorem

If f : R3 → R is a di�erentiable at a , then directional derivatives f ′ (a; t) exist

in all directions t and

f ′ (a; t) = ∇f (a) · t

Example

f (x , y) = x2 + y2, a = (1, 0), v = (x , y). Then Ta (v) = (2, 0) · v. And

∇f (a) = (2, 0)

If t = (1, 1) /√2, then

f ′ (a; t) = (2, 0) · (1, 1)/√2 =√2

Also note: ∣∣f ′ (a; t)∣∣ = |∇f (a)| |t| cosα ≤ |∇f (a)|I Change in f is maximum in the direction of gradient vector.


Theorem



f ′ (a; t) = ∇f (a) · t

Example


∇f (a) = (2, 0)

If t = (1, 1) /√2, then

f ′ (a; t) = (2, 0) · (1, 1)/√2 =√2



Theorem



f ′ (a; t) = ∇f (a) · t

Example


∇f (a) = (2, 0)

If t = (1, 1) /√2, then

f ′ (a; t) = (2, 0) · (1, 1)/√2 =√2



Geometric Interpretation

Gradient Vector is always perpendicular to level curves and level surfaces.

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

t

agradHfL

Along the tangent direction to alevel curve, directional derivativemust be 0! Function value does notchange in that direction. But then

f ′ (a; t) = ∇f (a) · t = 0




-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

t

agradHfL


f ′ (a; t) = ∇f (a) · t = 0




-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

t

agradHfL


f ′ (a; t) = ∇f (a) · t = 0


Extreme Values

∇f (a) = 0 =⇒ maximum or minimum or a saddle point.

x

y

z

-4 -2 0 2 4

-7.5

-5

-2.5

0

2.5

5

7.5


Extreme Values


x

y

z

-4 -2 0 2 4

-7.5

-5

-2.5

0

2.5

5

7.5


Extreme Values


x

y

z

-4 -2 0 2 4

-7.5

-5

-2.5

0

2.5

5

7.5


f and g are scalar �elds and c is a real constant

I ∇ (f + g) (a) = ∇f (a) +∇g(a)

I ∇ (cf ) = c∇fI ∇ (fg) = f∇g + g∇fI If r = (x , y , z) and r =

√x2 + y2 + z2, then ∇ (rn) = nrn−2r = nrn−1 r

I Chain Rule: f : R→ Rn and g : Rn → R and h = g ◦ f , thenh′(a) = ∇g · f ′(a)

Derivative of Vector Fields

De�nition

Let F : Rm → Rn be a vector �eld. Let a ∈ Rm be a point in space. F is said tobe di�erentiable at a if there exists a linear function Ta : Rm → Rn such that

F(a + v) = F (a) + Ta (v) + |v|Ra(v).



De�nition


F(a + v) = F (a) + Ta (v) + |v|Ra(v).


Linear Transformation m = n = 2

Let v = (x , y) ∈ R2. A linear transformation T : R2 → R2 can always bewritten as

T (v) =

[t11x + t12y

t21x + t22y

]=

[t11 t12t21 t22

] [x

y

]where t11, t12, t21, t22 ∈ R. That is, one can always �nd a matrix

t =

[t11 t12t21 t22

]such that

T (v) = t · v


De�nition


F(a + v) = F (a) + Ta (v) + |v|Ra(v).


If F : R2 → R2 and F(a) =

[Fx(a)Fy (a)

]then

Ta(v) =

[∂Fx∂x

∂Fx∂y

∂Fy∂x

∂Fy∂y

] [x

y

]


De�nition


F(a + v) = F (a) + Ta (v) + |v|Ra(v).


If F : R3 → R3 and F(a) =

Fx(a)Fy (a)Fz(a)

and v = (x , y , z) then

Ta(v) =

∂Fx∂x

∂Fx∂y

∂Fx∂z

∂Fy∂x

∂Fy∂y

∂Fy∂z

∂Fz∂x

∂Fz∂y

∂Fz∂z

x

y

z

Divergence and Curl of Vector Fields

Total derivative (which is like gradient) of a vector �eld is not very useful whenit comes to application is science and engineering.

De�nition

If A = (Ax ,Ay ,Az) is a di�erentiable vector �eld, then divergence of A(denotedby ∇ · A) is de�ned as

∇ · A =∂Ax

∂x+∂Ay

∂y+∂Az

∂z

De�nition

If A = (Ax ,Ay ,Az) is a di�erentiable vector �eld, then curl of A(denoted by∇× A) is de�ned as

∇× A = x

(∂Az

∂y− ∂Ay

∂z

)+ y

(∂Ax

∂z− ∂Az

∂x

)+ z

(∂Ay

∂x− ∂Ax

∂y

)



De�nition


∇ · A =∂Ax

∂x+∂Ay

∂y+∂Az

∂z

De�nition


∇× A = x

(∂Az

∂y− ∂Ay

∂z

)+ y

(∂Ax

∂z− ∂Az

∂x

)+ z

(∂Ay

∂x− ∂Ax

∂y

)



De�nition


∇ · A =∂Ax

∂x+∂Ay

∂y+∂Az

∂z

De�nition


∇× A = x

(∂Az

∂y− ∂Ay

∂z

)+ y

(∂Ax

∂z− ∂Az

∂x

)+ z

(∂Ay

∂x− ∂Ax

∂y

)

Gradient as an operator

It is useful to treat the symbol ∇ as a vector operator, that is when it operateson a scalar �eld and produces a vector �eld. It is conveniently written as

∇ = x∂

∂x+ y

∂

∂y+ z

∂

∂z

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